Understanding Time-Dependent Probability in Quantum Mechanics

[Irid]

Hi,
I'm a beginner at quantum mechanics and I wonder about this problem. Suppose I have found energy eigenstates of some potential, say, harmonic oscillator. Any state then can be expanded in terms of these eigenstates, and each term should be multiplied by the time factor exp(-iEt/h). What is the probability that the particle will be found in some particular eigenstate and how does it change in time? Using orthonormality of the energy eigenstates I find that the probability is just square of the amplitude and it doesn't change in time. But then, ANY state can be expanded in energy eigenstates, so this implies that the probability to be in ANY state is constant in time. Then, how can you change anything in quantum mechanics? Add/remove energy from the system, prepare the initial mix of states, change the mix?

 

[pam]

A new interaction added to the Hamiltonian can change the mix of states.
Otherwise, it is no different than normal modes in classical physics. The amplitude of each normal mode is also constant in time.

 

[CompuChip]

It's not strange; if the hamiltonian stays the same (e.g. the kinetic energy and the potential do not change) then it doesn't matter whether I measure now or in 20 minutes: chances of getting some particular result do not change. Only if I change something about the system (which reflects on the Hamiltonian, of course) the probabilities will change. Note however, that the wavefunction itself does change, it depends on time like
$$\Psi(x, t) = \sum_n c_n(t) \Psi_n(x)$$ with $$c_n(t) = e^{-i E_n t / \hbar} c_n(0)$$.

 

[comote]

Assuming the Hamiltonian is time independent then energy, and any observable that is compatible(commutes) with energy does not change over time in the absence of external influence.